Hmmm, i always said sky-rim and seer-uh-dill.
Makes me wonder about some of the others.
Elsweyr?
Is it pronounced els-were, els-where, else-where or els-way-er?
I always said els-were.
I'd say "Elsewhere". As the name comes from a Khajiit proverb:
As for other provinces I've always pronounced them like this:
Skyrim - Sky Rim
Cyrodiil - Seh Ro Dill
Morrowind - More O' Wind
Valenwood - Vay-Len Wood.
The rest are self explanatory really. High Rock, Hammerfell, Black Marsh, and Summset Isle.